Logistic Prey Growth

The first step to improve the biological properties of the L-V model was to modify the prey growth rate to include intraspecific competition for resources in the lower trophic level. In other words, the prey population is limited by the "struggle for existence" in a finite environment. This is usually done by substituting the logistic model, a(1 - H / K) for the prey growth function (aH) to give the predator-prey R-functions

We can see that this model staisfies Attribute 3, the decline in prey reproduction due to intraspecific competition for resources amongst prey. Unfortunately the prey model still fails Condition 2 because Attribute 4, consumer satiation, is not satisfied. In other words, even with logistic prey the L-V model still fails 4 of the 5 conditions for plausibility.

Population Dynamics

The zero-growth isocline for the prey equation is now given by

In other words, the prey zero-growth isocline begins at K on the H-axis when C = 0 and then slopes to the left as C gets larger until it intercepts the C-axis at a / b when H = 0 (see figure below). Notice that the predator isocline is not affected by the logistic modification to the prey model. Also notice that the model now damps to a stable equilibrium in a constant environment (no random variation due to weather, etc.).

  Figure. (Left) Herbivore (H) and carnivore (C) phase space showing zero-growth isoclines for the L-V model with intraspecific competition amongst prey (prey isocline = slanting blue line, predator isocline = vertical red line). (Right) Herbivore and carnivore time series plots showing damped cycles to equilibrium.